Linear Systems lecture 7 Fourier transforms academic year : 16-17 - - PowerPoint PPT Presentation
Linear Systems lecture 7 Fourier transforms academic year : 16-17 lecture : 7 UNIVERSITY OF TWENTE build : November 23, 2016 slides : 31 Today UNIVERSITY OF TWENTE The Fourier transform The Dirac delta function Properties of
UNIVERSITY OF TWENTE✳
academic year : 16-17 lecture : 7 build : November 23, 2016 slides : 31
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 1 intro
LS Today
NicoletTM iSTM50 Fourier Transform Infra Red Spectrometer
1 The Fourier transform 2 The fundamental theorem of Fourier transforms 3 Properties of the Fourier transform
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 2 1.1
LS The Fourier transform t x(t) x(t) x(t) The Fourier transform of a time-continuous, non-periodic signal x(t) is derived form the Fourier series of an approximating periodic signal.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 2 1.1
LS The Fourier transform
T 2
− T
2
t
T 2
− T
2
x(t) ˜ x(t) x(t) ˜ x(t) x(t) ˜ x(t) The Fourier transform of a time-continuous, non-periodic signal x(t) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x(t) by ˜ x(t) = x(t) if − T
2 < t < T 2 ,
where the period is T.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 2 1.1
LS The Fourier transform
T 2
− T
2
t x(t) ˜ x(t)
T 2
− T
2
x(t) ˜ x(t) x(t) ˜ x(t) The Fourier transform of a time-continuous, non-periodic signal x(t) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x(t) by ˜ x(t) = x(t) if − T
2 < t < T 2 ,
where the period is T. Let T → ∞, and analyse what happens with the Fourier series of ˜ x(t).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 2 1.1
LS The Fourier transform
T 2
− T
2
t x(t) ˜ x(t) x(t) ˜ x(t)
T 2
− T
2
x(t) ˜ x(t) The Fourier transform of a time-continuous, non-periodic signal x(t) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x(t) by ˜ x(t) = x(t) if − T
2 < t < T 2 ,
where the period is T. Let T → ∞, and analyse what happens with the Fourier series of ˜ x(t).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 3 1.2
LS From series to integral Let F be a function defined on the real numbers. Consider the sum
∞
F(n∆ω)∆ω.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 3 1.2
LS From series to integral Let F be a function defined on the real numbers. Consider the sum
∞
F(n∆ω)∆ω. ω
2∆ω 3∆ω 4∆ω 5∆ω 6∆ω ∆ω −∆ω −2∆ω
F ∆ω By regarding the sum as a Riemann sum, we see lim
∆ω→0+ ∞
F(n∆ω)∆ω =
∞
−∞
F(ω) dω.
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3
LS The Fourier integral Define ∆ω = 2π/T, then from the Fundamental theorem of Fourier series follows: ˜ x(t) =
∞
T
T/2
−T/2
x(τ)e−in∆ωτ dτ
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3
LS The Fourier integral Define ∆ω = 2π/T, then from the Fundamental theorem of Fourier series follows: ˜ x(t) =
∞
T
T/2
−T/2
x(τ)e−in∆ωτ dτ
= 1 2π
∞
T/2
−T/2
x(τ)ein∆ω(t−τ) dτ
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3
LS The Fourier integral Define ∆ω = 2π/T, then from the Fundamental theorem of Fourier series follows: ˜ x(t) =
∞
T
T/2
−T/2
x(τ)e−in∆ωτ dτ
= 1 2π
∞
T/2
−T/2
x(τ)ein∆ω(t−τ) dτ
T → ∞
T/2
−T/2
≈
∞
−∞
x(t) = 1 2π
∞
∞
−∞
x(τ)ein∆ω(t−τ) dτ
∆ω
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3
LS The Fourier integral Define ∆ω = 2π/T, then from the Fundamental theorem of Fourier series follows: ˜ x(t) =
∞
T
T/2
−T/2
x(τ)e−in∆ωτ dτ
= 1 2π
∞
T/2
−T/2
x(τ)ein∆ω(t−τ) dτ
T → ∞
T/2
−T/2
≈
∞
−∞
x(t) = 1 2π
∞
∞
−∞
x(τ)ein∆ω(t−τ) dτ
∆ω = 1 2π
∞
−∞
∞
−∞
x(τ)eiω(t−τ) dτ
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3
LS The Fourier integral Define ∆ω = 2π/T, then from the Fundamental theorem of Fourier series follows: ˜ x(t) =
∞
T
T/2
−T/2
x(τ)e−in∆ωτ dτ
= 1 2π
∞
T/2
−T/2
x(τ)ein∆ω(t−τ) dτ
T → ∞
T/2
−T/2
≈
∞
−∞
x(t) = 1 2π
∞
∞
−∞
x(τ)ein∆ω(t−τ) dτ
∆ω = 1 2π
∞
−∞
∞
−∞
x(τ)eiω(t−τ) dτ
= 1 2π
∞
−∞
∞
−∞
x(τ)e−iωτ dτ
eiωt dω
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 5 1.4
LS The Fourier transform
Definition
Let x(t) be a continuous-time signal. Then the Fourier transform of x(t) is defined as X(ω) =
∞
−∞
x(t)e−iωt dt, provided this integral exists.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 5 1.4
LS The Fourier transform
Definition
Let x(t) be a continuous-time signal. Then the Fourier transform of x(t) is defined as X(ω) =
∞
−∞
x(t)e−iωt dt, provided this integral exists. The Fourier transform X(ω) is sometimes called the spectrum of x.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 5 1.4
LS The Fourier transform
Definition
Let x(t) be a continuous-time signal. Then the Fourier transform of x(t) is defined as X(ω) =
∞
−∞
x(t)e−iωt dt, provided this integral exists. The Fourier transform X(ω) is sometimes called the spectrum of x. Convention: the name of the Fourier transform is the uppercase form of the of the signal.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 5 1.4
LS The Fourier transform
Definition
Let x(t) be a continuous-time signal. Then the Fourier transform of x(t) is defined as X(ω) =
∞
−∞
x(t)e−iωt dt, provided this integral exists. The Fourier transform X(ω) is sometimes called the spectrum of x. Convention: the name of the Fourier transform is the uppercase form of the of the signal. Alternative notation: X(ω) = F {x(t)} .
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 5 1.4
LS The Fourier transform
Definition
Let x(t) be a continuous-time signal. Then the Fourier transform of x(t) is defined as X(ω) =
∞
−∞
x(t)e−iωt dt, provided this integral exists. The Fourier transform X(ω) is sometimes called the spectrum of x. Convention: the name of the Fourier transform is the uppercase form of the of the signal. Alternative notation: X(ω) = F {x(t)} . If X(ω) is the Fourier transform of x(t), then we denote this as x(t) ↔ X(ω).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 6 1.5
LS The fundamental theorem of Fourier transforms
Theorem
Let x(t) be an absolutely integrable and piecewise smooth signal on R. If X(ω) is the Fourier transform of x(t), then 1 2π
∞
−∞
X(ω) eiωt dω = 1
2
(∗) where the integral is the Cauchy Prinicipal Value.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 6 1.5
LS The fundamental theorem of Fourier transforms
Theorem
Let x(t) be an absolutely integrable and piecewise smooth signal on R. If X(ω) is the Fourier transform of x(t), then 1 2π
∞
−∞
X(ω) eiωt dω = 1
2
(∗) where the integral is the Cauchy Prinicipal Value. Equation (∗) us also known as the Inversion Formula.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 6 1.5
LS The fundamental theorem of Fourier transforms
Theorem
Let x(t) be an absolutely integrable and piecewise smooth signal on R. If X(ω) is the Fourier transform of x(t), then 1 2π
∞
−∞
X(ω) eiωt dω = 1
2
(∗) where the integral is the Cauchy Prinicipal Value. Equation (∗) us also known as the Inversion Formula. The conditions imposed to x(t) are by no means necessary conditions: the theorem holds in many cases where x(t) is not absolutely integrable.
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 7 1.6
LS Fourier transforms vs series Fourier coefficients Fourier transform signal is periodic signal is non-periodic
cn = 1 T
x(t)e−inω0t dt X(ω) =
∞
−∞
x(t)e−iωt dt
n
cn
ω
X(ω) x(t) =
∞
cneinω0t x(t) = 1 2π
∞
−∞
X(ω)eiωt dω
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 8 1.7
LS Normalized Fourier coefficients
T 2
x(t)
− T
2
x(t) A signal x(t) has bounded support if there exists a number A > 0 such that x(t) = 0 for all |t| > A.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 8 1.7
LS Normalized Fourier coefficients
T 2
x(t)
− T
2
x(t) A signal x(t) has bounded support if there exists a number A > 0 such that x(t) = 0 for all |t| > A. Let ˜ x(t) be the periodic extension of x(t) on [−T/2, T/2]. Then for the Fourier coefficients cn of ˜ x we have cn = 1 T
T/2
−T/2
x(t)einω0t dt = ω0 2πX(nω0).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 8 1.7
LS Normalized Fourier coefficients
T 2
x(t)
− T
2
x(t) A signal x(t) has bounded support if there exists a number A > 0 such that x(t) = 0 for all |t| > A. Let ˜ x(t) be the periodic extension of x(t) on [−T/2, T/2]. Then for the Fourier coefficients cn of ˜ x we have cn = 1 T
T/2
−T/2
x(t)einω0t dt = ω0 2πX(nω0). Define the normalized Fourier coefficients: cn ω0 = 1 2πX(nω0)
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 9 1.8
LS Normalized Fourier coefficients For increasing values of T (and hence for decreasing values of ω0), the normalized Fourier coefficients fill up the Fourier transform X(ω).
ω
ω0
cn ω0
ω
ω0
cn ω0
ω
ω0
cn ω0
ω
1 2πX(ω)
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 10 1.9
LS The rectangular pulse
Example
Example 4.2.1
Find the Fourier transform
rect(t) 1
1 2
−1
2
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 10 1.9
LS The rectangular pulse
Example
Example 4.2.1
Find the Fourier transform
rect(t) 1
1 2
−1
2
X(ω) =
∞
−∞
rect(t)e−iωt dt =
1/2
−1/2
e−iωt dt.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 10 1.9
LS The rectangular pulse
Example
Example 4.2.1
Find the Fourier transform
rect(t) 1
1 2
−1
2
X(ω) =
∞
−∞
rect(t)e−iωt dt =
1/2
−1/2
e−iωt dt. If ω = 0 then X(0) =
1/2
−1/2
1 dt = 1 = Sa(0/2).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 10 1.9
LS The rectangular pulse
Example
Example 4.2.1
Find the Fourier transform
rect(t) 1
1 2
−1
2
X(ω) =
∞
−∞
rect(t)e−iωt dt =
1/2
−1/2
e−iωt dt. If ω = 0 then X(0) =
1/2
−1/2
1 dt = 1 = Sa(0/2). If ω = 0 then X(ω) =
1/2
−1/2
e−iωt dt = − 1 iω
−1/2
= − 1 iω
= 2 ω eiω/2 − e−iω/2 2i = 1 ω/2 sin(ω/2) = Sa(ω/2).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 10 1.9
LS The rectangular pulse
Example
Example 4.2.1
Find the Fourier transform
rect(t) 1
1 2
−1
2
X(ω) =
∞
−∞
rect(t)e−iωt dt =
1/2
−1/2
e−iωt dt. If ω = 0 then X(0) =
1/2
−1/2
1 dt = 1 = Sa(0/2). If ω = 0 then X(ω) =
1/2
−1/2
e−iωt dt = − 1 iω
−1/2
= − 1 iω
= 2 ω eiω/2 − e−iω/2 2i = 1 ω/2 sin(ω/2) = Sa(ω/2). For all ω ∈ R we have X(ω) = Sa(1
2ω).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 11 1.10
LS The rectangular pulse t
−3T/2 −T −T/2 T/2 T 3T/2 − 1
2 1 2
1
˜ x(t)
1 T
Let ˜ x(t) be the periodic extension of x(t) on
2 , T 2
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 11 1.10
LS The rectangular pulse t
−3T/2 −T −T/2 T/2 T 3T/2 − 1
2 1 2
1
˜ x(t)
1 T
Let ˜ x(t) be the periodic extension of x(t) on
2 , T 2
The duty cycle of ˜ x(t) is ρ = 1/T, and the normalized Fourier coefficients of ˜ x(t) are cn ω0 = ρ ω0 sinc(nρ) = 1 Tω0 Sa
nπ
T
2π Sa
2nω0
2πX(nω0).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 12 1.11
LS Existence of the Fourier transform
Definition
A function defined on a domain D is called absolutely integrable on D if
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 12 1.11
LS Existence of the Fourier transform
Definition
A function defined on a domain D is called absolutely integrable on D if
Theorem
If a signal x(t) is absolutely integrable on R then the Fourier transform X(ω) exists.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 12 1.11
LS Existence of the Fourier transform
Definition
A function defined on a domain D is called absolutely integrable on D if
Theorem
If a signal x(t) is absolutely integrable on R then the Fourier transform X(ω) exists. If x(t) is absolutely integrable then
a
x(t) dt
b
a
|x(t)| dt <
∞
−∞
|x(t)| dt < ∞, hence lim
a→−∞ lim b→∞
b
a
x(t) dt =
∞
−∞
x(t) dt exists.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 12 1.11
LS Existence of the Fourier transform
Definition
A function defined on a domain D is called absolutely integrable on D if
Theorem
If a signal x(t) is absolutely integrable on R then the Fourier transform X(ω) exists. If x(t) is absolutely integrable then
a
x(t) dt
b
a
|x(t)| dt <
∞
−∞
|x(t)| dt < ∞, hence lim
a→−∞ lim b→∞
b
a
x(t) dt =
∞
−∞
x(t) dt exists. If x(t) is absolutely integrable then X(ω) exists for all ω.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 13 1.12
LS One-sided exponential signals
Example
Example 4.2.3
Find the Fourier transform
t x(t)
1
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 13 1.12
LS One-sided exponential signals
Example
Example 4.2.3
Find the Fourier transform
t x(t)
1
Note that x(t) is absolutely integrable on R.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 13 1.12
LS One-sided exponential signals
Example
Example 4.2.3
Find the Fourier transform
t x(t)
1
Note that x(t) is absolutely integrable on R. For arbitrary ω we have X(ω) =
∞
−∞
e−αtu(t)e−iωt dt =
∞
e−(α+iω)t dt = lim
L→∞
L
e−(α+iω)t dt = lim
L→∞ −
1 α + iω e−(α+iω)t
= − 1 α + iω lim
L→∞
1 α + iω (0 − 1) = 1 α + iω .
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 13 1.12
LS One-sided exponential signals
Example
Example 4.2.3
Find the Fourier transform
t x(t)
1
Note that x(t) is absolutely integrable on R. For arbitrary ω we have X(ω) =
∞
−∞
e−αtu(t)e−iωt dt =
∞
e−(α+iω)t dt = lim
L→∞
L
e−(α+iω)t dt = lim
L→∞ −
1 α + iω e−(α+iω)t
= − 1 α + iω lim
L→∞
1 α + iω (0 − 1) = 1 α + iω .
For all α > 0 we have: e−αtu(t) ↔ 1 α + iω .
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 14 2.1
LS The spectrum of the Dirac delta function The spectrum of δ(t) can be found using the sifting property: F {δ(t)} =
∞
−∞
δ(t)e−iωt dt = e−iω0 = 1.
t
δ(t)
1
ω
F {δ(t)}
1
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 14 2.1
LS The spectrum of the Dirac delta function The spectrum of δ(t) can be found using the sifting property: F {δ(t)} =
∞
−∞
δ(t)e−iωt dt = e−iω0 = 1.
t
δ(t)
1
ω
F {δ(t)}
1
Is δ(t) = 1 2π
∞
−∞
F {δ(t)} eiωt dω?
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 14 2.1
LS The spectrum of the Dirac delta function The spectrum of δ(t) can be found using the sifting property: F {δ(t)} =
∞
−∞
δ(t)e−iωt dt = e−iω0 = 1.
t
δ(t)
1
ω
F {δ(t)}
1
Is δ(t) = 1 2π
∞
−∞
F {δ(t)} eiωt dω? The improper integral
∞
−∞
eiωt dω is not convergent:
∞
−∞
eiωt dω = lim
L→−∞ M→∞
M
L
eiωt dω = 1 it
M→∞ eiMt −
lim
L→−∞ eiLt
= ???
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 15 2.2
LS The Cauchy Principal Value
Definition
Let f (x) be a function defined on R. The Cauchy Principal Value of
∞
−∞
f (x) dx is defined as lim
L→∞
L
−L
f (x) dx.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 15 2.2
LS The Cauchy Principal Value
Definition
Let f (x) be a function defined on R. The Cauchy Principal Value of
∞
−∞
f (x) dx is defined as lim
L→∞
L
−L
f (x) dx. The integral I =
∞
−∞
x dx is not convergent.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 15 2.2
LS The Cauchy Principal Value
Definition
Let f (x) be a function defined on R. The Cauchy Principal Value of
∞
−∞
f (x) dx is defined as lim
L→∞
L
−L
f (x) dx. The integral I =
∞
−∞
x dx is not convergent. The Cauchy principal value of I is lim
L→∞
L
−L
x dx = lim
L→∞ 1 2L2 − 1 2(−L)2 = 0.
x
L −L
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 16 2.3
LS The Cauchy Principal Value for time-harmonic signals
Objective
Find the Cauchy principal value of
∞
−∞
eiωt dt for ω ∈ R.
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 16 2.3
LS The Cauchy Principal Value for time-harmonic signals
Objective
Find the Cauchy principal value of
∞
−∞
eiωt dt for ω ∈ R. Let L > 0, then for ω = 0:
L
−L
eiωt dt = 1 iω eiωt
−L = 1
iω
= 2 ω eiωL − e−iωL 2i = 2 ω sin(ωL) = 2L Sa(ωL).
UNIVERSITY OF TWENTE✳
The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 16 2.3
LS The Cauchy Principal Value for time-harmonic signals
Objective
Find the Cauchy principal value of
∞
−∞
eiωt dt for ω ∈ R. Let L > 0, then for ω = 0:
L
−L
eiωt dt = 1 iω eiωt
−L = 1
iω
= 2 ω eiωL − e−iωL 2i = 2 ω sin(ωL) = 2L Sa(ωL). If ω = 0 then
L
−L
eiωt dt =
L
−L
1 dt = 2L = 2L Sa(0 · L) = 2L Sa(ωL).
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LS The Cauchy Principal Value for time-harmonic signals
Objective
Find the Cauchy principal value of
∞
−∞
eiωt dt for ω ∈ R. Let L > 0, then for ω = 0:
L
−L
eiωt dt = 1 iω eiωt
−L = 1
iω
= 2 ω eiωL − e−iωL 2i = 2 ω sin(ωL) = 2L Sa(ωL). If ω = 0 then
L
−L
eiωt dt =
L
−L
1 dt = 2L = 2L Sa(0 · L) = 2L Sa(ωL). For all ω ∈ R:
L
−L
eiωt dt = 2L Sa(ωL)
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LS The Cauchy Principal Value for time-harmonic signals
Objective
Find the Cauchy principal value of
∞
−∞
eiωt dt for ω ∈ R. Let L > 0, then for ω = 0:
L
−L
eiωt dt = 1 iω eiωt
−L = 1
iω
= 2 ω eiωL − e−iωL 2i = 2 ω sin(ωL) = 2L Sa(ωL). If ω = 0 then
L
−L
eiωt dt =
L
−L
1 dt = 2L = 2L Sa(0 · L) = 2L Sa(ωL). For all ω ∈ R:
L
−L
eiωt dt = 2L Sa(ωL) Unfortunately lim
L→∞ 2L Sa(ωL) does not exist.
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LS The Cauchy Principal Value for time-harmonic signals As a function of ω, the graph of 2L Sa(ωL) is a narrow and high pulse:
ω
2L
2L Sa(ωL)
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 17 2.4
LS The Cauchy Principal Value for time-harmonic signals As a function of ω, the graph of 2L Sa(ωL) is a narrow and high pulse:
ω
2L
2L Sa(ωL)
The question is: can we use the sample function Sa(x) to emulate the Dirac delta function?
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 17 2.4
LS The Cauchy Principal Value for time-harmonic signals As a function of ω, the graph of 2L Sa(ωL) is a narrow and high pulse:
ω
2L
2L Sa(ωL)
The question is: can we use the sample function Sa(x) to emulate the Dirac delta function? Remember the definition of the delta function:
3. ∞
−∞
δ(x) dx = 1,
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 18 2.5
LS The Dirac delta function, once again Define pε(x) = 1 πx sin
x
ε
πε Sa
x
ε
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LS The Dirac delta function, once again Define pε(x) = 1 πx sin
x
ε
πε Sa
x
ε
pε(0) → ∞ for ε → 0+, if x = 0 then pε(x) = 0 for ε → 0+, ∞
−∞
pε(x) dx = 1 for all ε > 0, pε(x) is an even function.
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LS The Dirac delta function, once again Define pε(x) = 1 πx sin
x
ε
πε Sa
x
ε
pε(0) → ∞ for ε → 0+, if x = 0 then pε(x) = 0 for ε → 0+, ∞
−∞
pε(x) dx = 1 for all ε > 0, pε(x) is an even function.
If we want to use the sampling function to implement the delta function, we need to relax the second condition.
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LS The Dirac delta function, once again Define pε(x) = 1 πx sin
x
ε
πε Sa
x
ε
pε(0) → ∞ for ε → 0+, if x = 0 then pε(x) = 0 for ε → 0+, ∞
−∞
pε(x) dx = 1 for all ε > 0, pε(x) is an even function.
If we want to use the sampling function to implement the delta function, we need to relax the second condition. In stead of requiring that limε→0+ pε(x) = 0 we require lim
ε→0+
pε(x) dx = 0 for all δ > 0.
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LS The sine integral
Definition
The sine integral is the function Si(x) defined as Si(x) =
x
sin t t dt.
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LS The sine integral
Definition
The sine integral is the function Si(x) defined as Si(x) =
x
sin t t dt.
x π 2
−π
2
Si(x)
The sine integral is odd.
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LS The sine integral
Definition
The sine integral is the function Si(x) defined as Si(x) =
x
sin t t dt.
x π 2
−π
2
Si(x)
The sine integral is odd. lim
x→∞ Si(x) = π
2 and lim
x→−∞ Si(x) = −π
2
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LS The sine integral and the delta function Prove that for all δ > 0, that if ε → 0+ then
pε(x) dx =
−δ
−∞
pε(x) dx +
∞
δ
pε(x) dx → 0. Since pε is even, we just have to prove that the rightmost integral approaches 0.
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LS The sine integral and the delta function Prove that for all δ > 0, that if ε → 0+ then
pε(x) dx =
−δ
−∞
pε(x) dx +
∞
δ
pε(x) dx → 0. Since pε is even, we just have to prove that the rightmost integral approaches 0.
∞
δ
pε(x) dx = 1 πε
∞
δ
Sa
x
ε
y = x ε = 1 π
∞
δ/ε
Sa(y) dy = 1 π
ε
π
2 − Si
ε
2 − 1 π Si
ε
2 − 1 π · π 2 = 0 whenever ε → 0+.
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LS The Dirac delta function as a harmonic integral
Theorem
The Dirac delta function is defined as δ(x) = lim
ε→0+
1 πx sin
x
ε
ε→0+
1 πε Sa
x
ε
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LS The Dirac delta function as a harmonic integral
Theorem
The Dirac delta function is defined as δ(x) = lim
ε→0+
1 πx sin
x
ε
ε→0+
1 πε Sa
x
ε
Application
Let ω ∈ R, then
∞
−∞
eiωt dt = 2πδ(ω)
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LS The Dirac delta function as a harmonic integral
Theorem
The Dirac delta function is defined as δ(x) = lim
ε→0+
1 πx sin
x
ε
ε→0+
1 πε Sa
x
ε
Application
Let ω ∈ R, then
∞
−∞
eiωt dt = 2πδ(ω) Proof:
∞
−∞
eiωt dt = lim
L→∞ 2L Sa(ωL)
L = 1 ε = 2π lim
ε→0+
1 πε Sa(ω/ε) = 2πδ(ω).
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The Fourier transform The Dirac delta function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 21 2.8
LS The Dirac delta function as a harmonic integral
Theorem
The Dirac delta function is defined as δ(x) = lim
ε→0+
1 πx sin
x
ε
ε→0+
1 πε Sa
x
ε
Application
Let ω ∈ R, then
∞
−∞
eiωt dt = 2πδ(ω) Proof:
∞
−∞
eiωt dt = lim
L→∞ 2L Sa(ωL)
L = 1 ε = 2π lim
ε→0+
1 πε Sa(ω/ε) = 2πδ(ω). Corollary: 1 2π
∞
−∞
F {δ(t)} eiωt dt = δ(t).
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LS The Fourier transform of time-harmonic signals
Theorem
Example 4.2.8
For all α ∈ R: F
= 2π δ(ω − α).
ω
F
eiαt
2π α
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LS The Fourier transform of time-harmonic signals
Theorem
Example 4.2.8
For all α ∈ R: F
= 2π δ(ω − α).
ω
F
eiαt
2π α
Proof: F
=
∞
−∞
eiαte−iωt dt =
∞
−∞
ei(α−ω)t dt = 2π δ(α − ω) = 2π δ(ω − α).
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LS The Fourier transform of time-harmonic signals
Theorem
Example 4.2.8
For all α ∈ R: F
= 2π δ(ω − α).
ω
F
eiαt
2π α
Proof: F
=
∞
−∞
eiαte−iωt dt =
∞
−∞
ei(α−ω)t dt = 2π δ(α − ω) = 2π δ(ω − α). Special case: α = 0: 1 ↔ 2π δ(ω).
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LS Duality
Theorem
Section 3.4.8
Let x(t) be an absolutely integrable and piecewise smooth signal on R with Fourier transform X(ω), then X(t) ↔ 2π x(−ω).
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LS Duality
Theorem
Section 3.4.8
Let x(t) be an absolutely integrable and piecewise smooth signal on R with Fourier transform X(ω), then X(t) ↔ 2π x(−ω).
In this theorem the function X(ω) is regarded as a signal where ω is replaced by t.
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LS Duality
Theorem
Section 3.4.8
Let x(t) be an absolutely integrable and piecewise smooth signal on R with Fourier transform X(ω), then X(t) ↔ 2π x(−ω).
In this theorem the function X(ω) is regarded as a signal where ω is replaced by t. This theorem implies that functions and their Fourier transforms occur in pairs that exchange roles.
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LS Duality
Theorem
Section 3.4.8
Let x(t) be an absolutely integrable and piecewise smooth signal on R with Fourier transform X(ω), then X(t) ↔ 2π x(−ω).
In this theorem the function X(ω) is regarded as a signal where ω is replaced by t. This theorem implies that functions and their Fourier transforms occur in pairs that exchange roles. This phenomenon is called duality.
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LS Duality
Example
Find the Fourier transform of the sample function Sa(t).
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LS Duality
Example
Find the Fourier transform of the sample function Sa(t). We don’t want to compute the unpleasant integral F {Sa(t)} =
∞
−∞
sin t t e−iωt dt.
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LS Duality
Example
Find the Fourier transform of the sample function Sa(t). We don’t want to compute the unpleasant integral F {Sa(t)} =
∞
−∞
sin t t e−iωt dt. Note that rect(t/2) ↔ 2 Sa(ω) (see example 4.2.1).
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LS Duality
Example
Find the Fourier transform of the sample function Sa(t). We don’t want to compute the unpleasant integral F {Sa(t)} =
∞
−∞
sin t t e−iωt dt. Note that rect(t/2) ↔ 2 Sa(ω) (see example 4.2.1). From duality follows: 2 Sa(t) ↔ 2π rect(−ω/2) = 2π rect(ω/2).
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LS Duality
Example
Find the Fourier transform of the sample function Sa(t). We don’t want to compute the unpleasant integral F {Sa(t)} =
∞
−∞
sin t t e−iωt dt. Note that rect(t/2) ↔ 2 Sa(ω) (see example 4.2.1). From duality follows: 2 Sa(t) ↔ 2π rect(−ω/2) = 2π rect(ω/2). Hence F {Sa(t)} = π rect(ω/2).
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LS Linearity
Theorem
Section 4.3.1
Let x1(t) and x2(t) be time-continuous signals with Fourier transforms X1(ω) and X2(ω) respectively. Then for all α and β we have αx1(t) + βx2(t) ↔ αX1(ω) + βX2(ω).
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LS Linearity
Theorem
Section 4.3.1
Let x1(t) and x2(t) be time-continuous signals with Fourier transforms X1(ω) and X2(ω) respectively. Then for all α and β we have αx1(t) + βx2(t) ↔ αX1(ω) + βX2(ω). Write cos αt = eiαt + e−iαt 2 = 1
2eiαt + 1 2e−iαt, then
F {cos αt} = 1
2 · 2πδ(ω − α) + 1 2 · 2πδ(ω + α)
= π
see ex. 4.3.1
ω
F {cos αt} π α −α
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LS Real signals
Theorem
Section 4.3.2
Let x(t) be time-continuous signal with Fourier transform X(ω). If x(t) is a real signal then X(ω) = X(−ω).
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LS Real signals
Theorem
Section 4.3.2
Let x(t) be time-continuous signal with Fourier transform X(ω). If x(t) is a real signal then X(ω) = X(−ω). The Fourier transform of the one-sided exponential signal x(t) = e−αtu(t) (with α > 0) is X(ω) = 1 α + iω
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LS Real signals
Theorem
Section 4.3.2
Let x(t) be time-continuous signal with Fourier transform X(ω). If x(t) is a real signal then X(ω) = X(−ω). The Fourier transform of the one-sided exponential signal x(t) = e−αtu(t) (with α > 0) is X(ω) = 1 α + iω Observe that X(ω) = 1 α − iω = X(−ω).
R 1 α
C
ω = 0 X(ω) X(−ω)
ω → −∞ ω → +∞
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LS Time-reversal
Theorem
Let x(t) be time-continuous signal with Fourier transform X(ω), then F {x(−t)} = X(−ω).
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LS Time-reversal
Theorem
Let x(t) be time-continuous signal with Fourier transform X(ω), then F {x(−t)} = X(−ω). Example
t y(t) = e−α|t |
1
t x(t)=e−αtu(t)
1
The signal y(t) = e−α|t | can be written as y(t) = x(t) + x(−t), where x(t) = e−αtu(t) is the one-sided exponential.
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LS Time-reversal
Theorem
Let x(t) be time-continuous signal with Fourier transform X(ω), then F {x(−t)} = X(−ω). Example
t y(t) = e−α|t |
1
t x(t)=e−αtu(t)
1
The signal y(t) = e−α|t | can be written as y(t) = x(t) + x(−t), where x(t) = e−αtu(t) is the one-sided exponential. F {y(t)} = X(ω) + X(−ω) = 1 α + iω + 1 α − iω = 2α α2 + ω2 .
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LS Symmetry
Theorem
Let X(ω) be the Fourier transform of x(t), then if x(t) is even then X(ω) is even, if x(t) is odd then X(ω) is odd. If x(t) is real then if x(t) is even then X(ω) is real, if x(t) is odd then X(ω) is purely imaginary.
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LS Symmetry The signal cos αt is real and even.
ω
real axis
α −α π F {cos αt} = π
δ(ω + α) + δ(ω − α) is real and even.
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LS Symmetry The signal cos αt is real and even.
ω
real axis
α −α π F {cos αt} = π
δ(ω + α) + δ(ω − α) is real and even.
The signal sin αt is real and odd.
ω
imaginary axis
α −α iπ −iπ F {sin αt} = iπ
δ(ω + α) − δ(ω − α) is purely
imagenary and odd.
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LS Shifting and scaling in the time domain
Theorem
Section 4.3.3
Let x(t) be time-continuous signal with Fourier transform X(ω) and let α = 0, then F {x(t − t0)} = X(ω) e−iωt0.
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LS Shifting and scaling in the time domain
Theorem
Section 4.3.3
Let x(t) be time-continuous signal with Fourier transform X(ω) and let α = 0, then F {x(t − t0)} = X(ω) e−iωt0.
Theorem
Section 4.3.4
Let x(t) be time-continuous signal with Fourier transform X(ω) and let α = 0, then F {x(αt)} = 1 |α|X
ω
α
If α=−1 we get the time-reversal rule: x(−t)↔X(−ω).
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LS Even rectangular pulses
Example
Inspired by example 4.3.4
Let a > 0. Find the Fourier transform of x(t) = rect(t/a).
rect(t/a)
1
1 2a
−1
2a
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LS Even rectangular pulses
Example
Inspired by example 4.3.4
Let a > 0. Find the Fourier transform of x(t) = rect(t/a).
rect(t/a)
1
1 2a
−1
2a
Example 4.2.1: rect(t) ↔ Sa(ω/2).
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LS Even rectangular pulses
Example
Inspired by example 4.3.4
Let a > 0. Find the Fourier transform of x(t) = rect(t/a).
rect(t/a)
1
1 2a
−1
2a
Example 4.2.1: rect(t) ↔ Sa(ω/2). Scaling with factor α = 1/a gives F {x(t)} = a Sa(aω/2). ω a
4π a
2π a
−2π
a